1. Field of the Invention
The present invention relates generally to codewords useful for channel sounding and ranging in communication systems.
2. Description of the Related Art
Periodic Autocorrelation
It is well known by those skilled in the art, that codes which have so called perfect periodic autocorrelation are useful for channel sounding. If a code has perfect autocorrelation, then its periodic autocorrelation has one peak and is zero everywhere else.
Auto correlation is the cross correlation of a signal with itself. Say for example a code is of length N. Its periodic autocorrelation can be calculated by multiplying every element of it by every element of the same code, circularly shifted by a phase offset and summing the results of this multiplication. This is done at all possible phases.
The periodic autocorrelation R at lag j for signal xn isR(j)=Σnxnxn-j  [Eq. 1]
where Σn indicates the sum over all n.
Merit Factor
Another desirable property for a code is that it has low aperiodic autocorrelation side lobes. This is desirable especially in codes used to send data with Direct Sequence modulation. The lower the autocorrelation side lobes, the lower the amount of energy that spills over into adjacent symbols and so the lower the inter-symbol interference (ISI) produced by the codes themselves.
A commonly used measure of how good the code is in this respect is known as the Golay Merit Factor (GMF) or often simply the Merit Factor.
The Merit Factor is a measure of the quality of the autocorrelation function of a code. To calculate the merit factor the aperiodic autocorrelation function is calculated and then the merit factor is the ratio of the square of the central term to the sum of the squares of the off-peak terms.
This measure was introduced by M. J. E. Golay in “Sieves for Low Autocorrelation Binary Sequences”, IEEE Transactions on Information Theory, vol. IT-23, pp. 43-51, January 1977.
The binary sequences with the highest known Merit Factors are the length 13 and 11 Barker sequences with Merit factors of 14.08 and 12.10 respectively. By comparison, the mean Golay merit factor of the length 32 Walsh-Hadamard matrix is 0.194.
A code with perfect periodic autocorrelation, known here as a PAC code, is then a code where R(j) is zero for all j except when j=0.
A family of ternary sequences with perfect periodic autocorrelation was discovered by Valery Ipatov (“Ternary sequences with ideal autocorrelation properties”, Radio Eng. Electron. Phys., vol. 24, pp. 75-79, October 1979) and extended by T. Hoholdt, et al. (“Ternary sequences with Perfect Periodic Autocorrelation”, IEEE Transactions on Information Theory, vol. 29, no. 4, pp. 597-600, May 1983) (“Hoholdt, et al.”). There are many sequences in this family, for example, lengths 7, 13, 21, 31, 57, 63, 73, 91, 127, 133, 183, 273, 307, 381, 511, 553, 651, 757, 871, 993, 1057, 1407, 1723.
Cross Correlation
When a communications system 10 (see, FIG. 1) uses a set of codewords and the different codewords within that set are used for different purposes, e.g. to represent different data, or to provide isolation between different channels, it is desirable for these codewords to have low cross correlation with each other. This will reduce the chance of a receiver correlator being triggered by the wrong code. The wrong code could be present because, for example, there are echoes or reflections present of codes previously transmitted by transmitter 12, or for example, two or more channels are being received by receiver 14. Or the wrong code could be present because a different data code is being transmitted than is detected.
Spectral Peak to Average Ratio
For some communications systems the transmit power limits are set in terms of spectral power density. An example of this type of system is ultra wideband communications. The most useful part of the allowed transmit spectrum is flat. If a signal complies with these regulations, its worst case spectral density must stay below this straight line. If the signal spectrum has a peak anywhere, this peak must be kept below a certain limit by attenuating it and the rest of the signal. This effectively means that the whole signal must be penalized by any amount that the spectral peak to average ratio (SPAR) is above 0 dB. For this reason the SPAR needs to be kept as low as possible for codes used in these types of communications systems.
What is needed is a new technique which can generate codes with perfect periodic autocorrelation and, among other properties, relatively large Golay Merit Factors, good cross correlation and low spectral peak to average ratios.